**Shot noise** or **Poisson noise** is a type of noise which can be modeled by a Poisson process. In electronics shot noise originates from the discrete nature of electric charge. Shot noise also occurs in photon counting in optical devices, where shot noise is associated with the particle nature of light.

In a statistical experiment such as tossing a fair coin and counting the occurrences of heads and tails, the numbers of heads and tails after a great many throws will differ by only a tiny percentage, while after only a few throws outcomes with a significant excess of heads over tails or vice versa are common; if an experiment with a few throws is repeated over and over, the outcomes will fluctuate a lot. From the law of large numbers, one can show that the relative fluctuations reduce as the reciprocal square root of the number of throws, a result valid for all statistical fluctuations, including shot noise.

Shot noise exists because phenomena such as light and electric current consist of the movement of discrete (also called "quantized") 'packets'. Consider light—a stream of discrete photons—coming out of a laser pointer and hitting a wall to create a visible spot. The fundamental physical processes that govern light emission are such that these photons are emitted from the laser at random times; but the many billions of photons needed to create a spot are so many that the brightness, the number of photons per unit of time, varies only infinitesimally with time. However, if the laser brightness is reduced until only a handful of photons hit the wall every second, the relative fluctuations in number of photons, i.e., brightness, will be significant, just as when tossing a coin a few times. These fluctuations are shot noise.

The concept of shot noise was first introduced in 1918 by Walter Schottky who studied fluctuations of current in vacuum tubes.^{ [1] }

Shot noise may be dominant when the finite number of particles that carry energy (such as electrons in an electronic circuit or photons in an optical device) is sufficiently small so that uncertainties due to the Poisson distribution, which describes the occurrence of independent random events, are of significance. It is important in electronics, telecommunications, optical detection, and fundamental physics.

The term can also be used to describe any noise source, even if solely mathematical, of similar origin. For instance, particle simulations may produce a certain amount of "noise", where because of the small number of particles simulated, the simulation exhibits undue statistical fluctuations which don't reflect the real-world system. The magnitude of shot noise increases according to the square root of the expected number of events, such as the electric current or intensity of light. But since the strength of the signal itself increases more rapidly, the *relative* proportion of shot noise decreases and the signal-to-noise ratio (considering only shot noise) increases anyway. Thus shot noise is most frequently observed with small currents or low light intensities that have been amplified.

For large numbers, the Poisson distribution approaches a normal distribution about its mean, and the elementary events (photons, electrons, etc.) are no longer individually observed, typically making shot noise in actual observations indistinguishable from true Gaussian noise. Since the standard deviation of shot noise is equal to the square root of the average number of events *N*, the signal-to-noise ratio (SNR) is given by:

Thus when *N* is very large, the signal-to-noise ratio is very large as well, and any *relative* fluctuations in *N* due to other sources are more likely to dominate over shot noise. However, when the other noise source is at a fixed level, such as thermal noise, or grows slower than , increasing *N* (the DC current or light level, etc.) can lead to dominance of shot noise.

Shot noise in electronic circuits consists of random fluctuations of the electric current in a DC current which originate due to fact that current actually consists of a flow of discrete charges (electrons). Because the electron has such a tiny charge, however, shot noise is of relative insignificance in many (but not all) cases of electrical conduction. For instance 1 ampere of current consists of about 6.24×10^{18} electrons per second; even though this number will randomly vary by several billion in any given second, such a fluctuation is minuscule compared to the current itself. In addition, shot noise is often less significant as compared with two other noise sources in electronic circuits, flicker noise and Johnson–Nyquist noise. However, shot noise is temperature and frequency independent, in contrast to Johnson–Nyquist noise, which is proportional to temperature, and flicker noise, with the spectral density decreasing with increasing frequency. Therefore, at high frequencies and low temperatures shot noise may become the dominant source of noise.

With very small currents and considering shorter time scales (thus wider bandwidths) shot noise can be significant. For instance, a microwave circuit operates on time scales of less than a nanosecond and if we were to have a current of 16 nanoamperes that would amount to only 100 electrons passing every nanosecond. According to Poisson statistics the *actual* number of electrons in any nanosecond would vary by 10 electrons rms, so that one sixth of the time less than 90 electrons would pass a point and one sixth of the time more than 110 electrons would be counted in a nanosecond. Now with this small current viewed on this time scale, the shot noise amounts to 1/10 of the DC current itself.

The result by Schottky, based on the assumption that the statistics of electrons passage is Poissonian, reads^{ [2] } for the spectral noise density at the frequency ,

where is the electron charge, and is the average current of the electron stream. The noise spectral power is frequency independent, which means the noise is white. This can be combined with the Landauer formula, which relates the average current with the transmission eigenvalues of the contact through which the current is measured ( labels transport channels). In the simplest case, these transmission eigenvalues can be taken to be energy independent and so the Landauer formula is

where is the applied voltage. This provides for

commonly referred to as the Poisson value of shot noise, . This is a classical result in the sense that it does not take into account that electrons obey Fermi–Dirac statistics. The correct result takes into account the quantum statistics of electrons and reads (at zero temperature)

It was obtained in the 1990s by Khlus, Lesovik (independently the single-channel case), and Büttiker (multi-channel case).^{ [2] } This noise is white and is always suppressed with respect to the Poisson value. The degree of suppression, , is known as the Fano factor. Noises produced by different transport channels are independent. Fully open () and fully closed () channels produce no noise, since there are no irregularities in the electron stream.

At finite temperature, a closed expression for noise can be written as well.^{ [2] } It interpolates between shot noise (zero temperature) and Nyquist-Johnson noise (high temperature).

**Tunnel junction**is characterized by low transmission in all transport channels, therefore the electron flow is Poissonian, and the Fano factor equals one.**Quantum point contact**is characterized by an ideal transmission in all open channels, therefore it does not produce any noise, and the Fano factor equals zero. The exception is the step between plateaus, when one of the channels is partially open and produces noise.- A metallic diffusive wire has a Fano factor of 1/3 regardless of the geometry and the details of the material.
^{ [3] } - In 2DEG exhibiting fractional quantum Hall effect electric current is carried by quasiparticles moving at the sample edge whose charge is a rational fraction of the electron charge. The first direct measurement of their charge was through the shot noise in the current.
^{ [4] }

While this is the result when the electrons contributing to the current occur completely randomly, unaffected by each other, there are important cases in which these natural fluctuations are largely suppressed due to a charge build up. Take the previous example in which an average of 100 electrons go from point A to point B every nanosecond. During the first half of a nanosecond we would expect 50 electrons to arrive at point B on the average, but in a particular half nanosecond there might well be 60 electrons which arrive there. This will create a more negative electric charge at point B than average, and that extra charge will tend to *repel* the further flow of electrons from leaving point A during the remaining half nanosecond. Thus the net current integrated over a nanosecond will tend more to stay near its average value of 100 electrons rather than exhibiting the expected fluctuations (10 electrons rms) we calculated. This is the case in ordinary metallic wires and in metal film resistors, where shot noise is almost completely cancelled due to this anti-correlation between the motion of individual electrons, acting on each other through the coulomb force.

However this reduction in shot noise does not apply when the current results from random events at a potential barrier which all the electrons must overcome due to a random excitation, such as by thermal activation. This is the situation in p-n junctions, for instance.^{ [5] }^{ [6] } A semiconductor diode is thus commonly used as a noise source by passing a particular DC current through it.

In other situations interactions can lead to an enhancement of shot noise, which is the result of a super-poissonian statistics. For example, in a resonant tunneling diode the interplay of electrostatic interaction and of the density of states in the quantum well leads to a strong enhancement of shot noise when the device is biased in the negative differential resistance region of the current-voltage characteristics.^{ [7] }

Shot noise is distinct from voltage and current fluctuations expected in thermal equilibrium; this occurs without any applied DC voltage or current flowing. These fluctuations are known as Johnson–Nyquist noise or thermal noise and increase in proportion to the Kelvin temperature of any resistive component. However both are instances of white noise and thus cannot be distinguished simply by observing them even though their origins are quite dissimilar.

Since shot noise is a Poisson process due to the finite charge of an electron, one can compute the root mean square current fluctuations as being of a magnitude^{ [8] }

where *q* is the elementary charge of an electron, Δ*f* is the single-sided bandwidth in hertz over which the noise is considered, and *I* is the DC current flowing.

For a current of 100 mA, measuring the current noise over a bandwidth of 1 Hz, we obtain

If this noise current is fed through a resistor a noise voltage of

would be generated. Coupling this noise through a capacitor, one could supply a noise power of

to a matched load.

The flux signal that is incident on a detector is calculated as follows, in units of photons:

c is the speed of light, and h is the planck constant. Following Poisson statistics, the shot noise is calculated as the square root of the signal:

In optics, shot noise describes the fluctuations of the number of photons detected (or simply counted in the abstract) due to their occurrence independent of each other. This is therefore another consequence of discretization, in this case of the energy in the electromagnetic field in terms of photons. In the case of photon *detection*, the relevant process is the random conversion of photons into photo-electrons for instance, thus leading to a larger effective shot noise level when using a detector with a quantum efficiency below unity. Only in an exotic squeezed coherent state can the number of photons measured per unit time have fluctuations smaller than the square root of the expected number of photons counted in that period of time. Of course there are other mechanisms of noise in optical signals which often dwarf the contribution of shot noise. When these are absent, however, optical detection is said to be "photon noise limited" as only the shot noise (also known as "quantum noise" or "photon noise" in this context) remains.

Shot noise is easily observable in the case of photomultipliers and avalanche photodiodes used in the Geiger mode, where individual photon detections are observed. However the same noise source is present with higher light intensities measured by any photo detector, and is directly measurable when it dominates the noise of the subsequent electronic amplifier. Just as with other forms of shot noise, the fluctuations in a photo-current due to shot noise scale as the square-root of the average intensity:

The shot noise of a coherent optical beam (having no other noise sources) is a fundamental physical phenomenon, reflecting quantum fluctuations in the electromagnetic field. In optical homodyne detection, the shot noise in the photodetector can be attributed to either the zero point fluctuations of the quantised electromagnetic field, or to the discrete nature of the photon absorption process.^{ [9] } However, shot noise itself is not a distinctive feature of quantised field and can also be explained through semiclassical theory. What the semiclassical theory does not predict, however, is the squeezing of shot noise.^{ [10] } Shot noise also sets a lower bound on the noise introduced by quantum amplifiers which preserve the phase of an optical signal.

- Johnson–Nyquist noise or thermal noise
- 1/f noise
- Burst noise
- Contact resistance
- Image noise
- Quantum efficiency

**Analytical chemistry** studies and uses instruments and methods used to separate, identify, and quantify matter. In practice, separation, identification or quantification may constitute the entire analysis or be combined with another method. Separation isolates analytes. Qualitative analysis identifies analytes, while quantitative analysis determines the numerical amount or concentration.

**Stimulated emission** is the process by which an incoming photon of a specific frequency can interact with an excited atomic electron, causing it to drop to a lower energy level. The liberated energy transfers to the electromagnetic field, creating a new photon with a phase, frequency, polarization, and direction of travel that are all identical to the photons of the incident wave. This is in contrast to spontaneous emission, which occurs at a characteristic rate for each of the atoms/oscillators in the upper energy state regardless of the external electromagnetic field.

**Specific detectivity**, or * D**, for a photodetector is a figure of merit used to characterize performance, equal to the reciprocal of noise-equivalent power (NEP), normalized per square root of the sensor's area and frequency bandwidth.

In physics, a **phonon** is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. Often referred to as a quasiparticle, it is an excited state in the quantum mechanical quantization of the modes of vibrations for elastic structures of interacting particles. Phonons can be thought of as quantized sound waves, similar to photons as quantized light waves.

In quantum physics, a **quantum fluctuation** is the temporary random change in the amount of energy in a point in space, as prescribed by Werner Heisenberg's uncertainty principle. They are tiny random fluctuations in the values of the fields which represent elementary particles, such as electric and magnetic fields which represent the electromagnetic force carried by photons, W and Z fields which carry the weak force, and gluon fields which carry the strong force. Vacuum fluctuations appear as virtual particles, which are always created in particle-antiparticle pairs. Since they are created spontaneously without a source of energy, vacuum fluctuations and virtual particles are said to violate the conservation of energy. This is theoretically allowable because the particles annihilate each other within a time limit determined by the uncertainty principle so they are not directly observable. The uncertainty principle states the uncertainty in energy and time can be related by , where 1/2ħ ≈ 5,27286×10^{−35} Js. This means that pairs of virtual particles with energy and lifetime shorter than are continually created and annihilated in empty space. Although the particles are not directly detectable, the cumulative effects of these particles are measurable. For example, without quantum fluctuations the "bare" mass and charge of elementary particles would be infinite; from renormalization theory the shielding effect of the cloud of virtual particles is responsible for the finite mass and charge of elementary particles. Another consequence is the Casimir effect. One of the first observations which was evidence for vacuum fluctuations was the Lamb shift in hydrogen. In July 2020 scientists report that they, for the first time, measured that quantum vacuum fluctuations can influence the motion of macroscopic, human-scale objects by measuring correlations below the standard quantum limit between the position/momentum uncertainty of the mirrors of LIGO and the photon number/phase uncertainty of light that they reflect.

**Johnson–Nyquist noise** is the electronic noise generated by the thermal agitation of the charge carriers inside an electrical conductor at equilibrium, which happens regardless of any applied voltage. Thermal noise is present in all electrical circuits, and in sensitive electronic equipment such as radio receivers can drown out weak signals, and can be the limiting factor on sensitivity of an electrical measuring instrument. Thermal noise increases with temperature. Some sensitive electronic equipment such as radio telescope receivers are cooled to cryogenic temperatures to reduce thermal noise in their circuits. The generic, statistical physical derivation of this noise is called the fluctuation-dissipation theorem, where generalized impedance or generalized susceptibility is used to characterize the medium.

In physics, **screening** is the damping of electric fields caused by the presence of mobile charge carriers. It is an important part of the behavior of charge-carrying fluids, such as ionized gases, electrolytes, and charge carriers in electronic conductors . In a fluid, with a given permittivity *ε*, composed of electrically charged constituent particles, each pair of particles interact through the Coulomb force as

In physics, specifically in quantum mechanics, a **coherent state** is the specific quantum state of the quantum harmonic oscillator, often described as a state which has dynamics most closely resembling the oscillatory behavior of a classical harmonic oscillator. It was the first example of quantum dynamics when Erwin Schrödinger derived it in 1926, while searching for solutions of the Schrödinger equation that satisfy the correspondence principle. The quantum harmonic oscillator arise in the quantum theory of a wide range of physical systems. For instance, a coherent state describes the oscillating motion of a particle confined in a quadratic potential well. The coherent state describes a state in a system for which the ground-state wavepacket is displaced from the origin of the system. This state can be related to classical solutions by a particle oscillating with an amplitude equivalent to the displacement.

An **avalanche photodiode** (**APD**) is a highly sensitive semiconductor photodiode detector that exploits the photoelectric effect to convert light into electricity. From a functional standpoint, they can be regarded as the semiconductor analog of photomultipliers. The avalanche photodiode (APD) was invented by Japanese engineer Jun-ichi Nishizawa in 1952. However, study of avalanche breakdown, microplasma defects in Silicon and Germanium and the investigation of optical detection using p-n junctions predate this patent. Typical applications for APDs are laser rangefinders, long-range fiber-optic telecommunication, and quantum sensing for control algorithms. New applications include positron emission tomography and particle physics. APD arrays are becoming commercially available, also lightning detection and optical SETI may be future applications. It has been discovered in 2020 that adding graphene layer can prevent degradation over time to keep avalanche photodiodes *like new*, which is important in shrinking their size and costs for many diverse applications & bringing devices out of vacuum tubes into digital age.

A **photocathode** is a surface engineered to convert light (photons) into electrons using the photoelectric effect. Photocathodes are important in accelerator physics where they are used inside of a photoinjector to generate high brightness electron beams. Electron beams generated with photocathodes are commonly used for free electron lasers and for ultrafast electron diffraction. Photocathodes are also commonly used as the negatively charged electrode in a light detection device such as a photomultiplier or phototube.

The term **quantum efficiency** (**QE**) may apply to **incident photon to converted electron (IPCE) ratio** of a photosensitive device, or it may refer to the TMR effect of a Magnetic Tunnel Junction.

The **old quantum theory** is a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was rather a set of heuristic corrections to classical mechanics. The theory is now understood as the semi-classical approximation to modern quantum mechanics.

**Photon noise** is the randomness in signal associated with photons arriving at a detector. For a simple black body emitting on an absorber, the noise-equivalent power is given by

In physics, **quantum noise** refers to the uncertainty of a physical quantity that is due to its quantum origin. In certain situations, quantum noise appears as shot noise; for example, most optical communications use amplitude modulation, and thus, the quantum noise appears as shot noise only. For the case of uncertainty in the electric field in some lasers, the quantum noise is not just shot noise; uncertainties of both amplitude and phase contribute to the quantum noise. This issue becomes important in the case of noise of a quantum amplifier, which preserves the phase. The phase noise becomes important when the energy of the frequency modulation or phase modulation of waves is comparable to the energy of the signal.

**Image noise** is random variation of brightness or color information in images, and is usually an aspect of electronic noise. It can be produced by the image sensor and circuitry of a scanner or digital camera. Image noise can also originate in film grain and in the unavoidable shot noise of an ideal photon detector. Image noise is an undesirable by-product of image capture that obscures the desired information.

In electronics, **noise** is an unwanted disturbance in an electrical signal. Noise generated by electronic devices varies greatly as it is produced by several different effects.

In probability theory and statistics, the **Poisson distribution**, named after French mathematician Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.

**Phonon noise**, also known as **thermal fluctuation noise**, arises from the random exchange of energy between a thermal mass and its surrounding environment. This energy is quantized in the form of phonons. Each phonon has an energy of order , where is Boltzmann's constant and is the temperature. The random exchange of energy leads to fluctuations in temperature. This occurs even when the thermal mass and the environment are in thermal equilibrium, i.e. at the same time-average temperature. If a device has a temperature-dependent electrical resistance, then these fluctuations in temperature lead to fluctuations in resistance. Examples of devices where phonon noise is important include bolometers and calorimeters. The superconducting transition edge sensor (TES), which can be operated either as a bolometer or a calorimeter, is an example of a device for which phonon noise can significantly contribute to the total noise.

**Photon statistics** is the theoretical and experimental study of the statistical distributions produced in photon counting experiments, which use Photodetectors to analyze the intrinsic statistical nature of photons in a light source. In these experiments, light incident on the photodetector generates photoelectrons and a counter registers electrical pulses generating a statistical distribution of photon counts. Low intensity disparate light sources can be differentiated by the corresponding statistical distributions produced in the detection process.

**Super-resolution photoacoustic imaging** is a set of techniques used to enhance spatial resolution in photoacoustic imaging. Specifically, these techniques primarily break the optical diffraction limit of the photoacoustic imaging system. It can be achieved in a variety of mechanisms, such as blind structured illumination, multi-speckle illumination, or photo-imprint photoacoustic microscopy in Figure 1.

- ↑ Schottky, W. (1918). "Über spontane Stromschwankungen in verschiedenen Elektrizitätsleitern".
*Annalen der Physik*(in German).**362**(23): 541–567. Bibcode:1918AnP...362..541S. doi:10.1002/andp.19183622304. English translation in: On spontaneous current fluctuations in various electrical conductors - 1 2 3 Blanter, Ya. M.; Büttiker, M. (2000). "Shot noise in mesoscopic conductors".
*Physics Reports*. Dordrecht: Elsevier.**336**(1–2): 1–166. arXiv: cond-mat/9910158 . Bibcode:2000PhR...336....1B. doi:10.1016/S0370-1573(99)00123-4. S2CID 119432033. - ↑ Beenakker, C.W.J.; Büttiker, M. (1992). "Suppression of shot noise in metallic diffusive conductors" (PDF).
*Physical Review B*.**46**(3): 1889–1892. Bibcode:1992PhRvB..46.1889B. doi:10.1103/PhysRevB.46.1889. hdl: 1887/1116 . PMID 10003850. - ↑ V.J. Goldman, B. Su (1995). "Resonant Tunneling in the Quantum Hall Regime: Measurement of Fractional Charge".
*Science*.**267**(5200): 1010–1012. Bibcode:1995Sci...267.1010G. doi:10.1126/science.267.5200.1010. PMID 17811442. S2CID 45371551. See also Description on the researcher's website Archived 2008-08-28 at the Wayback Machine . - ↑ Horowitz, Paul and Winfield Hill, The Art of Electronics, 2nd edition. Cambridge (UK): Cambridge University Press, 1989, pp. 431–2.
- ↑ Bryant, James, Analog Dialog, issue 24-3
- ↑ Iannaccone, Giuseppe (1998). "Enhanced Shot Noise in Resonant Tunneling: Theory and Experiment".
*Physical Review Letters*.**80**(5): 1054–1057. arXiv: cond-mat/9709277 . Bibcode:1998PhRvL..80.1054I. doi:10.1103/physrevlett.80.1054. S2CID 52992294. - ↑ Thermal and Shot Noise. Appendix C. Retrieved from class notes of Prof. Cristofolinini, University of Parma. Archived on Wayback Machine. [url=https://web.archive.org/web/20181024162550/http://www.fis.unipr.it/~gigi/dida/strumentazione/harvard_noise.pdf]
- ↑ Carmichael, H. J. (1987-10-01). "Spectrum of squeezing and photocurrent shot noise: a normally ordered treatment".
*JOSA B*.**4**(10): 1588–1603. Bibcode:1987JOSAB...4.1588C. doi:10.1364/JOSAB.4.001588. ISSN 1520-8540. - ↑ Leonard., Mandel (1995).
*Optical coherence and quantum optics*. Wolf, Emil. Cambridge: Cambridge University Press. ISBN 9780521417112. OCLC 855969014.

- This article incorporates public domain material from the General Services Administration document: "Federal Standard 1037C".(in support of MIL-STD-188)

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